InfoSci®-Journals Annual Subscription Price for New Customers: As Low As US$ 4,950

This collection of over 175 e-journals offers unlimited access to highly-cited, forward-thinking content in full-text PDF and XML with no DRM. There are no platform or maintenance fees and a guarantee of no more than 5% increase annually.

Receive the complimentary e-books for the first, second, and third editions with the purchase of the Encyclopedia of Information Science and Technology, Fourth Edition e-book. Plus, take 20% off when purchasing directly through IGI Global's Online Bookstore.

Take 20% Off All Publications Purchased Directly Through the IGI Global Online Bookstore: www.igi-global.com/

Abstract

Engineering phenomena described through models come with uncertainties arising out of the modelling assumptions and observational statistical discrepancies. A probabilistic framework helps in quantifying the uncertainties involved. Conventional statistics are not very useful for any inferring from a limited amount of data which is often the case for many real life engineering problem. Bayesian inference tackles this issue by integrating past experience with current information. This chapter introduces the basic concepts of Bayesian inference and illustrates through three problems in structural engineering, how it can be useful. The emphasis is in showing how Markov chain Monte Carlo (MCMC) simulation is used in Bayesian inference.

Introduction

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain they do not refer to reality. — Albert Einstein

Structural engineering problems are usually analysed with the aid of models to predict the behaviour of structures under consideration. These models can be defined either in terms of derived physics based approaches or probabilistic approaches (Der Kiureghian & Ditlevsen, 2009). Subjected to environmental loads, a structure’s performance is usually measured in terms of stresses induced, deformations, deterioration levels, and other damage measures. Prediction of these quantities, by adopting either of these modelling approaches, results in an improper estimate as the models themselves may contain some uncertainty. There are various sources of uncertainty which need to be taken into account for a better prediction of response quantities. Based on the available literature (Pate-Cornell, 1996; Der Kiureghian & Ditlevsen, 2009; Kennedy et al., 1980; Kennedy & Ravindra, 1984) uncertainties are characterised in two basic categories viz. aleatory and epistemic. Aleatory uncertainty includes intrinsic randomness in predicting the outcome of a system. This type of uncertainty always exists as there is inherent natural randomness in any system. This randomness can very well be explained by an example of a concrete cube compressive test. No two test results of a concrete cube test (with the same target ) are the same, because of the inherent randomness that exists in the material and other properties. Epistemic uncertainty is due to a lack of knowledge of the behaviour of the structure being studied/analysed. These may be due to the differences in modelling approach for a single system. Inclusion of these uncertain quantities in prediction model will result in an accurate probabilistic estimation of structural responses.

Usually, we adopt simplified models based on our (prior) beliefs and assumptions within the accepted principles of mechanics. These assumptions include over simplification of mathematical models, conversion of mathematical expressions into computational models (e.g., Finite Element (FE) models) and knowledge acquired from limited experiments. Such models limit our capability of predicting the actual behaviour of a structure. Additionally, observations are typically replete with errors due to imperfections in measurements (especially non-destructive measurements), difficulty in acquiring data (Oden et al., 2010) and lack of understanding about the behaviour of the structure. Moreover, due to limited availability of these data, a proper inference of the system may lack clarity.

Probabilistic assessment of a structure requires the model to be defined in the probabilistic domain. Simulation methods, for example Monte Carlo simulation (MCS), Latin hypercube sampling (LHS), importance sampling and subset simulation (SS), rely on the existing models to accurately predict the desired response. This prediction does not take into account the information/data available through newly performed testing/measurements. Bayesian inference is one such method which can incorporate a newly acquired information to update the existing model. Conventional statistics relies on a large amount of data to take safety related decisions. In the presence of a limited number of real observations available for a structure, pursuing the Bayesian inference is apt for the treatment of uncertainties (Ang & Tang, 2007). The Bayesian approach facilitates the integration of multiple prediction models, various sources of uncertainty, errors, and inspection data, towards overall uncertainty quantification for the engineering system/phenomenon under consideration. In the present work, Bayesian updating is used for incorporating uncertainties occurring in structural engineering systems.